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1051
TIFR2021-Maths-A: 16
The matrix $\begin{pmatrix} 4 & -3 & -3\\3 & -2 & -3\\ -1 & 1& 2 \end{pmatrix}$ is diagonalizable nilpotent idempotent none of the other three options
The matrix$$\begin{pmatrix} 4 & -3 & -3\\3 & -2 & -3\\ -1 & 1& 2 \end{pmatrix}$$isdiagonalizablenilpotentidempotentnone of the other three options
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Sep 27, 2021
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1052
TIFR2021-Maths-A: 17
Which of the following is a necessary and sufficient condition for two real $3\times 3$ matrices $A$ and $B$ to be similar $($i.e., $PAP^{-1}=B$ for an invertible real $3\times 3$ matrix $P)$? They have the same characteristic polynomial They have the same minimal polynomial They have the same minimal and characteristic polynomials None of the other three conditions
Which of the following is a necessary and sufficient condition for two real $3\times 3$ matrices $A$ and $B$ to be similar $($i.e., $PAP^{-1}=B$ for an invertible real $3...
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1053
TIFR2021-Maths-A: 18
Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition: $A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degree at most } 2\}, \text{ and} $ ... $[B:A]$ of $A$ in $B$ equals $1$ $2$ $4$ none of the other three options
Consider the following two subgroups $A,B$ of the group $\mathbb{Q}[x]$ of one variable rational polynomials under addition:$$A=\{p(x)\in \mathbb{Z}[x]|p \text{ has degre...
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120
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Sep 27, 2021
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1054
TIFR2021-Maths-A: 19
Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection? $43$ $45$ $47$ none of the other three options
Let $G$ be any finite group of order $2021$. For which of the following positive integers $m$ is the map $G\rightarrow G$, given by $g\mapsto g^m$, a bijection?$43$$45$$4...
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229
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Sep 27, 2021
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1055
TIFR2021-Maths-A: 20
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have? $13$ $16$ $4$ $25$
How many subgroups does $(\mathbb{Z}/13\mathbb{Z})\times (\mathbb{Z}/13\mathbb{Z})$ have?$13$$16$$4$$25$
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Sep 27, 2021
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1056
TIFR2021-Maths-B: 1
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2 dx=0,$ then $\displaystyle\lim_{n\rightarrow \infty} f_n\left(\frac{1}{2}\right)=0.$
Let $f_n:[0,1]\rightarrow \mathbb{R}$ be a continuous function for each positive integer $n$. If $$\displaystyle\lim_{n\rightarrow \infty} \displaystyle \int_0^1 f_n(x)^2...
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166
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Sep 27, 2021
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1057
TIFR2021-Maths-B: 2
Let $(X,d)$ be an infinite compact metric space. Then there exists no function $f:X\rightarrow X$, continuous or otherwise, with the property that $d(f(x),f(y))>d(x,y)$ for all $x\neq y$.
Let $(X,d)$ be an infinite compact metric space. Then there exists no function $f:X\rightarrow X$, continuous or otherwise, with the property that $d(f(x),f(y))>d(x,y)$ f...
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141
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Sep 27, 2021
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1058
TIFR2021-Maths-B: 3
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
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1059
TIFR2021-Maths-B: 4
If $X$ is a compact metric space, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.
If $X$ is a compact metric space, there exists a surjective (not necessarily continuous) function $\mathbb{R}\rightarrow X$.
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152
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1060
TIFR2021-Maths-B: 5
If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.
If $X$ is a compact metric space, then every isometry $f:X\rightarrow X$ is surjective.
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120
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1061
TIFR2021-Maths-B: 6
Define a metric on the set of finite subsets of $\mathbb{Z}$ as ollows: $d(A,B)=\text{the cardinality of } (A\cup B \backslash (A\cap B)).$ The resulting metric space admits an isometry into $\mathbb{R}^n,$ for some positive integer $n$.
Define a metric on the set of finite subsets of $\mathbb{Z}$ as ollows:$$d(A,B)=\text{the cardinality of } (A\cup B \backslash (A\cap B)).$$The resulting metric space adm...
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102
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1062
TIFR2021-Maths-B: 7
There exists a continuous function $f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$ such that $f(0)=0$ and $f(1)=\text{Id}$.
There exists a continuous function$$f:[0,1]\rightarrow \{A\in M_2(\mathbb{R})|A^2=A\}$$such that $f(0)=0$ and $f(1)=\text{Id}$.
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113
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1063
TIFR2021-Maths-B: 8
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such that $f(x)=x$.
Let $f:[0,1]\rightarrow{\mathbb{R}}$ be a monotone increasing (not necessarily continuous) function such that $f(0)>0$ and $f(1)<1$. Then there exists $x\in[0,1]$ such th...
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113
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Sep 27, 2021
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1064
TIFR2021-Maths-B: 9
The set $\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$ is finite.
The set$$\{(x,y)\in \mathbb{N}\times\mathbb{N}| x^y \text{ divides } y^x,\:x\neq y,\:xy\neq0,\:x\neq1\}$$is finite.
soujanyareddy13
96
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1065
TIFR2021-Maths-B: 10
Suppose a line segment of a fixed length $L$ is given. It is possible to construct a triangle of perimeter $L$, whose angles are $105^{\circ},\: 45^{\circ} \text{ and } 30^{\circ}$, using only a straight edge and a compass.
Suppose a line segment of a fixed length $L$ is given. It is possible to construct a triangle of perimeter $L$, whose angles are $105^{\circ},\: 45^{\circ} \text{ and } 3...
soujanyareddy13
130
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Sep 27, 2021
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1066
TIFR2021-Maths-B: 11
The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n$.
The real vector space $M_n(\mathbb{R})$ cannot be spanned by nilpotent matrices, for any positive integer $n$.
soujanyareddy13
150
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Sep 27, 2021
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1067
TIFR2021-Maths-B: 12
Let $S \subseteq M_n(\mathbb{R})$ be a nonempty finite set closed under matrix multiplication. Then there exists $A\in S$ such that the trace of $A$ is an integer.
Let $S \subseteq M_n(\mathbb{R})$ be a nonempty finite set closed under matrix multiplication. Then there exists $A\in S$ such that the trace of $A$ is an integer.
soujanyareddy13
119
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1068
TIFR2021-Maths-B: 13
Given a linear transformation $T:\mathbb{Q}^4\rightarrow\mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V)\underline\subset V.$
Given a linear transformation $T:\mathbb{Q}^4\rightarrow\mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V)\underline\subset V.$
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85
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Sep 27, 2021
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1069
TIFR2021-Maths-B: 14
If G is a finite group such that the group $\text{Aut}(G)$ of automorphisms of $G$ is cyclic, then $G$ is abelian.
If G is a finite group such that the group $\text{Aut}(G)$ of automorphisms of $G$ is cyclic, then $G$ is abelian.
soujanyareddy13
83
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Sep 27, 2021
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1070
TIFR2021-Maths-B: 15
There exists a countable group having uncountably many subgroups.
There exists a countable group having uncountably many subgroups.
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111
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Sep 27, 2021
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